Ordinarization numbers of numerical semigroups

TL;DR

This paper investigates the enumeration of numerical semigroups based on their ordinarization number, using geometric and combinatorial methods to derive formulas and analyze properties for different classes of semigroups.

Contribution

It introduces a geometric approach to count numerical semigroups with fixed ordinarization numbers and extends known results to new classes of semigroups.

Findings

Derived a formula for semigroups with ordinarization number 2.

Connected the problem to counting integer points in rational polyhedral cones.

Analyzed ordinarization numbers for semigroups generated by two elements and intervals.

Abstract

There has been significant recent interest in studying how the number of numerical semigroups of genus $g$ behaves as a function of $g$. Bras-Amor\'os has shown how to organize the collection of numerical semigroups of genus $g$ into a rooted tree called the ordinarization tree. The ordinarization number of a numerical semigroup $S$ is the length of the path from $S$ back to the root of the tree. We study the problem of counting numerical semigroups of genus $g$ with a fixed ordinarization number $r$. We show how this can be interpreted as a counting problem about integer points in a certain rational polyhedral cone and use ideas from Ehrhart theory to study this problem. We give a formula for the number of numerical semigroups of genus $g$ and ordinarization number $2$, building on the corresponding result of Bras-Amor\'os for ordinarization number $1$. We show that the ordinarization number of a numerical semigroup generated by two elements is equal to the number of integer points in a certain right triangle with rational vertices. We consider the analogous problem for supersymmetric numerical semigroups with more generators. We also study ordinarization numbers of numerical semigroups generated by an interval.